arX

iv:g

r-qc

/990

7040

v1 9

Jul

199

9

On Applications of Campbell’s Embedding Theorem

James E. Lidsey1a, Carlos Romero2b, Reza Tavakol1c

& Steve Rippl1∗

1 School of Mathematical Sciences

Queen Mary & Westfield CollegeMile End Road

London E1 4NS, UK

2 Departamento de Fısica

Universidade Federal da ParaıbaC. Postal 5008 - J. Pessoa -Pb

58059-970 - Brazil

February 7, 2008

Abstract

A little known theorem due to Campbell [1] is employed to establish the localembedding of a wide class of 4–dimensional spacetimes in 5–dimensional Ricci–flatspaces. An embedding for the class of n–dimensional Einstein spaces is also found.The local nature of Campbell’s theorem is highlighted by studying the embeddingof some lower–dimensional spaces.

1 Introduction

There has been considerable interest in recent years in theories of gravity that containa different number of spatial dimensions from the usual three in general relativity (GR).One physical motivation for considering more than three spatial dimensions arises fromthe Kaluza–Klein interpretation of the fundamental interactions [2, 3, 4, 5]. Extra spatialdimensions also arise naturally in supergravity [4] and superstring theories [6] and mayhave played an important role in the evolution of the very early Universe [3, 7].

A new version of 5–dimensional GR has recently been developed by Wesson andothers [8, 9, 10]. In this approach, the energy density and pressure of the (3 + 1)–dimensional energy–momentum tensor arise directly from the extra components of the(4 + 1)–dimensional Einstein tensor, (5)Gab, where it is assumed that (5)Gab = 0. Thus,

∗e-mail: (a) jel@maths.qmw.ac.uk, (b) cromero@dfjp.ufpb.br, (c) reza@maths.qmw.ac.uk

the physics of (3 + 1)–dimensional cosmologies may be recovered, in principle, from thegeometry of (4 + 1)–dimensional, vacuum GR [9]. Direct calculations have verified thatthe spatially flat, perfect fluid cosmologies may be derived in this way [9, 10].

Lower–dimensional theories of gravity have also been extensively studied in recentyears [11]. These theories are interesting because they may provide a solvable frameworkwithin which many of the technical and conceptual problems associated with quantumgravitational effects in (3 + 1) dimensions may be addressed. A question that naturallyarises in these studies, however, is the extent to which the results and intuitions obtainedin lower dimensions may be directly carried over to the (3 + 1)–dimensional environmentand vice versa [12, 13]. In general, the precise relationship between these theories and(3 + 1)–dimensional Einstein gravity is not clear. For example, GR does not exhibit aNewtonian limit in (2 + 1) dimensions [14], whereas other theories, such as a modifiedversion of the Brans–Dicke theory, do have such a limit [13].

An investigation into how lower– and higher–dimensional theories of gravity are re-lated to 4–dimensional GR is therefore well motivated from a physical point of view. Apotential bridge between gravitational theories of different dimensionality may be foundby employing the embedding relationships that exist between spaces and the main purposeof this paper is to investigate such relationships further.

The embedding of manifolds in higher dimensions is also interesting from a purelymathematical point of view. For example, it allows an alternative, invariant classificationof known solutions to Einstein’s field equations to be made [15]. Furthermore, the em-bedding method may lead to new solutions. Indeed, the maximal analytic extension ofthe Schwarzschild solution was independently found in this way [16].

A number of embedding theorems are in existence. It is well known that an analyticLorentzian space Vn(s, t), with s spacelike and t timelike dimensions, where n = s + t,can be locally and isometrically embedded into a higher dimensional, pseudo–euclideanspace EN(S, T ), where N = S + T , n ≤ N ≤ n(n + 1)/2, and S ≥ s and T ≥ tare positive integers [18]. The line element of EN is given by ds2 = eA(dxA)2, whereA = (0, 1, . . . , N − 1) and eA = ±1. Thus, no more than ten dimensions are requiredto embed all 4–dimensional solutions to Einstein’s field equations. On the other hand,if the Ricci tensor of Vn(s, t) is zero, then N ≥ n + 2 [19]. This implies that no curved,4–dimensional solutions to the vacuum Einstein equations can be locally embedded in a5–dimensional flat space; the minimal embedding space is E6 [20].

There also exists a theorem due to Campbell [1, 23]:

Theorem: Any analytic Riemannian space Vn(s, t) can be locally embedded in a Ricci–

flat, Riemannian space Vn+1(s, t), where s = s and t = t+ 1, or s = s+ 1 and t = t.

Campbell’s theorem implies that all solutions to the n–dimensional Einstein field equa-tions with arbitrary energy–momentum tensor can be embedded, at least locally, in aspacetime that is itself a solution to (n+ 1)—dimensional, vacuum GR [25]. (We refer toa local embedding in the usual differential geometric sense, i.e., without any direct refer-ence to the global topology of the embedding or the embedded spaces). This theorem istherefore closely related to Wesson’s procedure [8], since it implies that any 4–dimensionalcosmology may be locally embedded in a 5–dimensional Ricci–flat space, at least in prin-ciple. Very little reference to Campbell’s theorem can be found in the literature (see,however, Magaard [23] and Goenner [24]). Recently, Romero, Tavakol and Zalaletdinov

[25] outlined the proof of this theorem in a modern notation and discussed its relationshipwith Wesson’s procedure [8]. They also emphasised its constructive nature with the helpof some concrete examples.

Here we investigate some further applications of Campbell’s theorem. Section 2 sum-marizes the main points of the theorem. In Sections 3 and 4 we consider the embedding ofthe general class of 4–dimensional spacetimes that admit a non–twisting null Killing vec-tor [15]. In Section 5 we find an embedding space for the general class of n–dimensionalEinstein spaces. We then proceed in Section 6 to discuss some aspects of Campbell’stheorem regarding the local and global embeddings of lower–dimensional gravity. Weconclude in Section 7.

2 The Embedding Theorem of Campbell

We begin by discussing the embedding theorem due to Campbell [1, 23, 25]. Consider thespace Vn(s, t) with metric (n)gαβ(x

µ) and line element1

(n)ds2 = (n)gαβ(xµ)dxαdxβ , (2.1)

and let the local embedding of this space in the manifold Vn+1(s, t) be given by

(n+1)ds2 = gαβ(xµ, ψ)dxαdxβ + ǫφ2(xµ, ψ)dψ2, (2.2)

where ǫ = ±1 and ψ is the coordinate that spans the extra dimension. It is assumed thatgαβ, when restricted to a hypersurface ψ = ψ0, results in (n)gαβ:

gαβ(xµ, ψ0) = (n)gαβ(x

µ). (2.3)

According to Campbell’s theorem [1], the functional form of the higher–dimensionalmetric coefficients (2.2) can be determined if functions Ωαβ(x

µ, ψ) may be found whichsatisfy the set of conditions

Ωαβ = Ωβα (2.4)

Ωαβ;α = Ω,β (2.5)

ΩαβΩαβ − Ω2 = −ǫ (n)R (2.6)

on some hypersurface ψ = ψ0 and if the functions gαβ and Ωαβ evolve in accordance withthe equations

∂gαβ∂ψ

= −2φΩαβ , (2.7)

∂Ωαβ

∂ψ= φ

(

−ǫ(n)Rαβ + ΩΩα

β

)

+ ǫgαλφ;λβ, (2.8)

1In this paper, Greek indices take values in the range (0, 1, . . . , n − 1), Latin indices run from(0, 1, . . . , n), semicolons and commas indicate covariant and partial differentiations respectively and space-time metrics have signature (+,−,−, . . .).

respectively2. In these expressions, Ωαβ ≡ (n)gαλΩλβ , Ω ≡ (n)gαβΩαβ and (n)R ≡

(n)Rµν(n)gµν .

If Eqs. (2.7) and (2.8) are evaluated on the hypersurface ψ = ψ0, it can be shown thatEqs. (2.4)–(2.8) are equivalent to the vacuum, (n + 1)–dimensional GR field equations(n+1)Rab(x

µ, ψ0) = 0 [25]. Moreover, it can be proved that Eqs. (2.4)–(2.6) are valid for allψ in the neighbourhood of ψ0 when Eqs. (2.7) and (2.8) are satisfied [1]. This implies thatthe Ricci tensor of Vn+1 vanishes for any ψ in the neighbourhood of ψ0. Consequently,Eq. (2.2) may be viewed as an embedding of the metric (2.1) in a Ricci–flat, (n + 1)–dimensional space.

Applications of Campbell’s theorem considered in this paper are based up on theintegrability of Eq. (2.8). We consider the embedding of spaces with vanishing Ricciscalar curvature ((n)R = 0) and also find an embedding for the class of Einstein spaces((n)R = constant). In the former case, one solution to Eqs. (2.4)–(2.6) is given byΩαβ = 0. It should be emphasized, however, that this does not necessarily represent themost general solution possible. Thus, the embedding of spaces with (n)R = 0 may bedivided into two subclasses. These correspond to embeddings where all the componentsof Ωαβ vanish and to those where some (or all) of the components are non–trivial.

As an example of an application of this theorem, we will conclude this Section bydiscussing the subclass of embeddings where Ωαβ = 0. It follows from Eq. (2.7) that∂gαβ/∂ψ = 0 and gαβ is therefore independent of the extra coordinate ψ. This impliesthat gαβ = (n)gαβ in the neighbourhood of the hypersurface ψ = ψ0. The one remainingequation that needs to be solved is Eq. (2.8) which simplifies to

(n)gαλφ;λβ = (n)Rαβφ. (2.9)

We conclude, therefore, that the embedding metric is given by

(n+1)ds2 = (n)gαβdxαdxβ + ǫφ2dψ2, (2.10)

where φ is a solution to Eq. (2.9).Taking the trace of Eq. (2.9) implies that φ must satisfy the massless Klein–Gordon

equation, (n)gαβφ;αβ = 0. If we assume an embedding of this form, therefore, a necessary,but not sufficient, condition on φ is that it be an harmonic function of xµ. This restrictionoften provides valuable insight into the generic form that φ must take if it is to satisfythe full set of differential equations (2.9). We remark that similar conclusions hold whenVn is Ricci–flat. Indeed, one solution to Eq. (2.9) in this case is φ = 1 and this providesa simple proof of theorem III of Romero et al. [25].

An interesting consequence of this embedding is that it may be repeated indefi-nitely, at least in principle. That is, the Ricci–flat space with metric (2.10) may itselfbe embedded in an (n + 2)–dimensional, Ricci–flat space with a line–element given by(n+2)ds2 = (n+1)ds2 + ϕ2dθ2, where θ represents the extra coordinate and ϕ = ϕ(xµ, ψ, θ)

2We note here that with φ = 1, n = 3, there is a clear parallel with the language used in the 1 + 3decomposition employed in the ADM formalism [21] and the initial value formulation of GR [22]. Thiscan be seen through the following identifications:

t→ ψ, hαβ → (3)gαβ, Kαβ → Ωαβ ,

where hαβ is the metric of the 3-space and Kαβ is the extrinsic curvature.

is an harmonic function satisfying (n+1)gαλϕ;λβ = 0. Thus, once a given Lorentzian spaceVn has been embedded in a Ricci–flat space Vn+1, further embeddings in Ricci–flat spacesof progressively higher dimensions can be considered.

Having summarized the steps that need to be taken when applying Campbell’s the-orem, we proceed in the following Sections to investigate the local embedding of a wideclass of Lorentzian spaces with one timelike dimension. We begin by considering the em-bedding of 4–dimensional spacetimes that admit a non–twisting null Killing vector fieldk, where k(µ;ν) = 0, kµk

µ = 0 and k[µkν;ρ] = 0. It can be shown that there are two classesof metrics that admit a Killing vector of this form [15], depending up on whether k iscovariantly constant, kµ;ν = 0, or whether it satisfies the less severe restriction k(µ;ν) = 0.

3 The Embedding of Spacetimes Admitting a Covari-

antly Constant Null Killing Vector Field

Metrics admitting a covariantly constant, null Killing vector field k have the form

ds2 = dudv + fdu2 − dx2 − dy2, (3.1)

where kµ = ∂µu and f = f(u, x, y) is an arbitrary function that is independent of thecoordinate v [15]. The coordinates (x, y) span the spacelike 2–surfaces that are orthogonalto k and the surfaces u = constant are null. The Ricci scalar for these spacetimesvanishes for arbitrary f , whilst the Riemann and Ricci tensors are given by Rµνρσ =−2k[µ∂ν]∂[ρfkσ] and Rµν = 1

2(∂2T f)kµkν , respectively, where ∂2

T is the Laplacian on thetransverse 2–surfaces. The only non–zero components of these tensors are

Ruxux = −1

2f,xx, Ruxuy = −1

2f,xy, Ruyuy = −1

2f,yy (3.2)

and

Ruu =1

2(f,xx + f,yy) (3.3)

and it follows from Eq. (3.2) that linear terms in f of the form a(u)+bi(u)xi do not affect

the Riemann tensor. They can therefore be transformed away.The class of spacetimes given by Eq. (3.1) is physically very interesting. They are

known as plane waves when f(u, xi) = hij(u)xixj for some symmetric function hij(u).

These solutions with hii = 0 were first discussed by Brinkman [31]. A purely gravitationalwave is characterized by the condition hii(u) = 0 and a purely electromagnetic wavecorresponds to hij(u) = h(u)δij, where h(u) ≥ 0 [28]. The amplitudes of the gravitationaland electromagnetic waves are given by the trace–free part of hij and by [Tr(hij)]

1/2,respectively. In general, the amplitudes may be arbitrary functions of u.

When f is a solution to the Laplace equation ∂2Tf = 0, the manifold is Ricci–flat. In

this case, Eq. (3.1) represents the most general, 4–dimensional solution to the vacuumEinstein field equations with a covariantly constant null vector [15]. The dependenceof f on u may be arbitrary and many different solutions can therefore be considered.More general solutions to Laplace’s equation, where hij also depends on xk, are known asplane–fronted waves.

Plane–fronted waves are solutions to any gravitational theory whose field equationsare given in terms of a second–rank tensor derived from the curvature tensor and itsderivatives [30]. This property may be traced to the fact that the curvature is null.Included in this class of theories is string theory [32]. Indeed, plane–fronted waves areexact solutions to the classical equations of motion to all orders in σ–model perturbationtheory [29, 30]. Exact solutions that include non–trivial dilaton and antisymmetric tensorfields can also be found and correspond to the case where ∂2

T f is an arbitrary function ofu [30].

Since R = 0 for all f , we may begin by choosing Ωαβ = 0. An embedding metricis therefore given by Eq. (2.10), where φ = φ(xµ, ψ) satisfies Eq. (2.9). This equationrepresents the set of coupled differential equations:

f,xφ,v − φ,ux = 0 (3.4)

f,yφ,v − φ,uy = 0 (3.5)

− 2f,uφ,v − f,xφ,x − f,yφ,y + 2φ,uu = (f,xx + f,yy)φ (3.6)

φ,uv = φ,vv = φ,vx = φ,vy = φ,xx = φ,xy = φ,yy = 0. (3.7)

Differentiating Eqs. (3.4) and (3.5) both with respect to x and y implies that

f,xxφ,v = f,xyφ,v = f,yyφ,v = 0, (3.8)

where we have employed Eq. (3.7). It follows from Eqs. (3.2) and (3.8) that the Riemanncurvature tensor must vanish if φ,v 6= 0, thereby implying that the spacetime is flat.Consequently, φ must be independent of v when Ωαβ = 0. In this case, Eqs. (3.4)–(3.6)simplify to

φ,ux = 0 (3.9)

φ,uy = 0 (3.10)

− f,xφ,x − f,yφ,y + 2φ,uu = (f,xx + f,yy)φ. (3.11)

The embedding metric is determined once Eqs. (3.9)–(3.11) have been solved subjectto the constraints (3.7). We will now consider the vacuum and non–vacuum cases in turn.

3.1 The Embedding of Vacuum Plane–fronted Waves

The right–hand side of Eq. (3.11) vanishes for the vacuum solutions. Differentiating thisequation with respect to both x and y then implies that

f,xxφ,x + f,xyφ,y = 0

f,xyφ,x + f,yyφ,y = 0 (3.12)

and combining these two equations implies that

f,xx(

φ2,x + φ2

,y

)

= 0

f,xy(

φ2,x + φ2

,y

)

= 0. (3.13)

If the embedded spacetime is vacuum and has non-zero curvature, the second derivativesof f with respect to x and y must be non–vanishing. Consequently, Eq. (3.13) can only be

satisfied if φ2,x = −φ2

,y. However, φ should be a real function if the embedding spacetimeis to be physical. Thus, φ must be independent of both x and y.

The only non–trivial constraint that remains in Eqs. (3.9)–(3.11), therefore, is thatφ,uu = 0 and this has the general solution φ = a(ψ) + b(ψ)u, where a and b are arbitraryfunctions of the fifth coordinate ψ. One possible embedding of 4–dimensional, vacuum,plane–fronted waves in a 5–dimensional, Ricci–flat manifold is therefore given by

ds2 = dudv + fdu2 − dx2 − dy2 − (a(ψ) + b(ψ)u)2 dψ2. (3.14)

A second local embedding of these plane–fronted waves can be found by assuming that

Ωαβ =

f/(2ψ0) if α = β = u0 otherwise

(3.15)

on the hypersurface ψ = ψ0. In this case, the only non–zero components of Ωαβ and Ωαβ

are Ωvu = 2Ωuu and Ωvv = 4Ωuu, respectively, where indices have been raised with (4)gαβ.

It follows immediately that Ω = ΩαβΩαβ = 0, so Eq. (2.6) is satisfied. Moreover, Eq.

(2.5) simplifies to Ωvu;v = 0. Since both Ωα

β and (4)gαβ are independent of v, however,this condition is also satisfied.

Eq. (2.8) is solved by specifying φ = −1, since Ωαβ is independent of ψ. Thus, the

solution to Eq. (2.7) that satisfies the initial conditions (2.3) on the hypersurface ψ = ψ0

is given by

gαβ =

(ψ/ψ0)f if α = β = u(4)gαβ otherwise.

(3.16)

It follows, therefore, that when indices are raised with gαβ, the only non–zero componentsof Ωα

β and Ωαβ are Ωvu and Ωvv, as before. Thus, Eqs. (2.4) and (2.6) are valid for

arbitrary ψ. The same conclusion holds for Eq. (2.5), since gαβ is itself independent of v.We may conclude, therefore, that the 5–dimensional embedding for this ansatz is given

by

ds2 = dudv +ψ

ψ0fdu2 − dx2 − dy2 − dψ2. (3.17)

It may be verified by direct calculation that this space is Ricci-flat. Its curvature differsfrom that of Eq. (3.14), however. In particular, we find that Ruxuψ = −(1/2ψ0)f,x andRuyuψ = −(1/2ψ0)f,y, whereas these components vanish for the spacetime correspondingto Eq. (3.14).

3.2 The Embedding of Electromagnetic Waves and Exact String

Backgrounds

We may consider the more general class of spacetimes characterized by

f(u, xk) = hij(u)xixj + fT (u, xk), (3.18)

where fT is an arbitrary solution to ∂2T fT = 0. These spacetimes are not Ricci–flat if

Tr(hij) 6= 0, since ∂2T f = 2(h11 + h22).

Now, the general form of φ consistent with Eqs. (3.7), (3.9) and (3.10) is given by

φ = a(u) + bx+ cy, (3.19)

where a(u) is an arbitrary function of u and b and c are arbitrary constants. (We assumefor simplicity that φ is independent of the fifth coordinate). The embedding spacetimemay therefore be determined by finding a solution to Eq. (3.11) that is consistent withEq. (3.19).

Let us begin with the simpler case where b = c = 0, so that φ is a function only of u.Substitution of Eqs. (3.18) and (3.19) into Eq. (3.11) then implies that

d2a

du2= (h11 + h22) a. (3.20)

In this case, the embedding metric is given by

ds2 = dudv + fdu2 − dx2 − dy2 − a2(u)dψ2 (3.21)

and may be expressed in a closed form whenever an exact solution to Eq. (3.20) canbe found for a given hij(u). This embedding is general, in the sense that the amplitudehij is an arbitrary function of u. The functional form of φ is also independent of fT , sowe may consider arbitrary forms for this latter function, subject to the condition that itsatisfies the Laplace equation ∂2

TfT = 0. Eq. (3.22) corresponds to the embedding of anelectromagnetic plane wave of arbitrary amplitude when h12 = fT = 0 and h11 = h22. Inthis case, a space with vanishing Weyl tensor is embedded in a space with vanishing Riccitensor of the form

ds2 = dudv +

(

1

2a

d2a

du2

)

(

x2 + y2)

du2 − dx2 − dy2 − a2(u)dψ2. (3.22)

An embedding is also possible if (b, c) 6= 0 and fT = 0. In this case, Eq. (3.20) stillapplies, but the components of hij are restricted by the additional constraints

(2h22 + h11)1/2 (2h11 + h22)

1/2 = ∓h12

b = ±(

2h22 + h11

2h11 + h22

)1/2

c. (3.23)

We find that the embedding metric is given by

ds2 = dudv + hijxixjdu2 − dx2 − dy2 − [a(u) + bx+ cy]2 dψ2 (3.24)

when Eqs. (3.20) and (3.23) are satisfied.

4 The Embedding for Spacetimes Admitting a Non–

Constant Null Killing Vector Field

If the null Killing vector is not (covariantly) constant, the metric is given by

ds2 = 2xdu(dv +mdu) − x−1/2(dx2 + dy2), (4.1)

where m = m(u, x, y) is an arbitrary function and is independent of v [15]. The Riccicurvature scalar of these spacetimes vanishes for arbitrary m and the only non–trivialcomponent of the Ricci tensor is given by

Ruu = x1/2(

(xm,x),x + xm,yy

)

. (4.2)

We will consider the subset of spacetimes that are solutions to vacuum GR. This includesa wide class of spaces, since the dependence ofm on u is arbitrary. When m is independentof u and Ruu = 0 the spacetimes (4.1) are the stationary van Stockum solutions [15].

Following the discussion of Section 2, we choose Ωαβ = 0, since the Ricci scalar is zero.The (u, u), (u, y), (x, u), (x, v), (x, y) and (y, y) components of Eq. (2.9) are then givenby

x1/2φ,x − 2φ,uv = 0 (4.3)

φ,vy = 0 (4.4)

2xm,xφ,v − 2xφ,ux + φ,u = 0 (4.5)

2xφ,vx − φ,v = 0 (4.6)

4xφ,xy + φ,y = 0 (4.7)

4xφ,yy − φ,x = 0, (4.8)

respectively. However, differentiation of Eq. (4.8) with respect to v implies that φ,vx = 0,where we have employed Eq. (4.4). Thus, Eq. (4.6) implies that φ must be independentof v, but it then follows from Eq. (4.3) that φ must also be independent of x. Eq. (4.7)then implies that φ must also be independent of y and, finally, Eq. (4.5) implies that φ isindependent of u. In conclusion, therefore, the only consistent solution is that φ = φ(ψ)when Ωαβ = 0.

An alternative embedding may be found by assuming the ansatz

Ωαβ =

xm/ψ0 if α = β = u0 otherwise.

(4.9)

The argument is similar to that followed in the previous Section. Eqs. (2.4)–(2.6) aresatisfied on a particular hypersurface ψ = ψ0, because (4)guu = 0 and (4)gαβ and Ωαβ areindependent of v. One solution to Eq. (2.8) is given by φ = −1 and this implies that theintegral of Eq. (2.7) is given by

gαβ =

2xm(ψ/ψ0) if α = β = u(4)gαβ otherwise.

(4.10)

The embedding metric is therefore given by

ds2 = 2xdudv + 2xm

(

ψ

ψ0

)

du2 − x−1/2dx2 − x−1/2dy2 − dψ2. (4.11)

It may be verified that Eqs. (2.4)–(2.6) remain valid when indices are raised with gαβ, sothey are valid for all ψ.

To summarize thus far, we have found embedding spaces for the general class of metricsthat admit a non–twisting, null Killing vector field. All these spacetimes have vanishingcurvature scalar, however. We therefore extend our analysis in the following Section toinclude the class of spacetimes for which the curvature scalar is covariantly constant.

5 The Embedding of Einstein spaces

The class of n–dimensional Einstein spaces is defined by

(n)Rαβ =

(

κ

n

)

(n)δαβ,(n)R = κ, (5.1)

where κ is an arbitrary constant [33]. For n ≥ 3, we may define Λ ≡ (2 − n)κ/(2n).This may be viewed as a cosmological constant in the vacuum Einstein field equations(n)Gα

β = Λ(n)δαβ.We proceed by specifying φ = 1. Eq. (2.8) then reduces to

∂Ωαβ

∂ψ=

−ǫκnδαβ + ΩΩα

β. (5.2)

This equation admits the exact solution

Ωαβ = aψ−1δαβ (5.3)

on the specific hypersurface

ψ = ψ0 = ±(

an(1 + an)

ǫκ

)1/2

, (5.4)

where a is a constant. When ǫ = −1 (corresponding to an extra spacelike coordinate),we require for consistency that −1/n < a < 0 if κ > 0 and a < −1/n or a > 0 if κ < 0.Conversely, for ǫ = +1, we require a > 0 or a < −1/n for κ > 0 and −1/n < a < 0 ifκ < 0.

We will assume for the moment that the solution (5.3) is valid for arbitrary values of ψwhen (n)Rα

β is calculated with gαβ(xµ, ψ) rather than with the original metric (n)gαβ(x

µ).(The validity of this assumption will be verified shortly for a = −1). We may nowintegrate Eq. (2.7) subject to the initial conditions (2.3). We find that

gαβ(xµ, ψ) = ψ−2a

(

an(1 + an)

ǫκ

)a(n)gαβ(x

µ). (5.5)

The functions Ωαβ are then determined by this equation and Eq. (5.3). The result is

Ωαβ = aψ−1−2a

(

an(1 + an)

ǫκ

)a(n)gαβ. (5.6)

However, Eqs. (2.4)—(2.6) must also be satisfied. The functions Ωαβ are symmetric,so Eq. (2.4) is clearly valid. Moreover, they are independent of xµ, so Eq. (2.5) is alsoconsistent. On the other hand, Eq. (2.6) is more restrictive because Ω = anψ−1 6= 0 ingeneral. Indeed, this equation is only satisfied if a = −1. Thus, the solution (5.3) onlyapplies if ǫκ > 0, which implies that ǫ = −1 (ǫ = +1) for a positive (negative) Λ.

Finally, Eq. (2.8) must be considered when (n)Rαβ is calculated with the n–dimensional

part of the (n+ 1)–dimensional embedding metric. This will establish the validity of thisembedding procedure. For a given metric gαβ, we may perform the conformal transfor-mation gαβ = kgαβ, where k is constant. It follows that Rα

β = k−1Rαβ and R = k−1R.

Using this property, we are able to calculate (n)Rαβ with the metric gαβ(x

µ, ψ) = k(n)gαβ,since k = ǫκψ2/[n(n− 1)] is a constant. We find that

(n)Rαβ(x

µ, ψ) = k−1Rαβ(x

µ) =

[

n− 1

ǫψ2

]

(n)δαβ (5.7)

and one can further show by an analogous argument that

(n)R(xµ, ψ) = k−1R =n(n− 1)

ǫψ2. (5.8)

Direct substitution of Eqs. (5.7) and (5.8) then implies that Eqs. (2.6) and (2.8) are validfor arbitrary ψ. Consequently, the embedding metric is locally Ricci–flat for every valueof ψ, as required, and it is given by

(n+1)ds2 =

[

ǫκ

n(n− 1)ψ2

]

(n)gαβ(xµ)dxαdxβ + ǫdψ2. (5.9)

The embeddings that we have considered in this paper thus far are local, in the sensethat no reference was made to the global topology of either the embedded or embeddingspace. This is because Campbell’s theorem is a local theorem. In the next Section, weshall highlight the local nature of this theorem further by investigating the embeddingsof some lower–dimensional spacetimes.

6 Local and Global Embedding of Spaces with Lower

Dimensions

Clarke [34] has proved that any C∞-Riemannian manifold Vn with Ck-Riemannian metric(k ≥ 3) of rank r and signature s can be globally Ck-isometrically embedded in Em(p, q),where

m = p+ q, p ≥ n− r + s

2+ 1 (6.1)

andq ≥ n

2(3n+ 11) (6.2)

if Vn is compact and

q ≥ n

2(2n2 + 27) +

5

2n2 + 1 (6.3)

if Vn is non–compact. Clearly, an analogue of this theorem is required, where the embed-ding space is Ricci, rather than Riemann, flat. Unfortunately, such a theorem does not asyet exist, but the lower bounds (6.2) and (6.3) suggest that more than one extra dimen-sion may generally be needed for global embeddings in Ricci–flat spaces. Nevertheless,some aspects of Campbell’s theorem with regard to local and global embeddings can behighlighted by considering lower–dimensional examples.

6.1 Embedding of (1 + 1)–dimensional Spaces

Campbell’s theorem implies that any (1 + 1)–dimensional space can be locally embeddedin a 3–dimensional, Ricci–flat space M3. However, the Weyl tensor vanishes identicallyin three dimensions, so the embedding space is necessarily flat, i.e., (3)Rµνλρ ≡ 0. Thus,Campbell’s theorem is equivalent to Friedman’s theorem in this case [18].

This would seem to imply that Campbell’s theorem results in a trivial embedding of all(1+1)–dimensional spaces. It is important to emphasize, however, that the topology of theembedding space is not specified in this procedure, due to the local nature of the theorem.In principle, therefore, a given space Vn may be (locally) embedded into more than onehigher–dimensional, Ricci–flat space, each of which has a different global topology. Thisfollows since there is usually more than one solution to Eqs. (2.7) and (2.8) consistentwith the boundary conditions (2.3). Moreover, the range of the extra coordinate ψ is notspecified in Campbell’s approach. It may be either compact or non–compact and this willalso affect the topology of the embedding space. Within the context of (1+1) dimensions,this implies that the metric on M3 may not cover the whole of Minkowski space. It ispossible, therefore, that the embedding space may contain singularities and, indeed, itmay even exhibit a non–trivial causal structure.

These features may be illustrated by considering different embeddings of the (1 + 1)–dimensional Minkowski space (2)dη2 = dt2 − dx2, where −∞ ≤ t, x ≤ +∞. It followsfrom Section 2 that one class of embedding metrics is given by (3)ds2 = (2)dη2 − φ2dψ2,where3 ∂µ∂νφ = 0. The general solution to these equations is given by φ = aµ(ψ)xµ,where aµ are arbitrary functions.

The whole of M3 is covered if φ = 1 and −∞ ≤ ψ ≤ +∞. However, a non–trivialembedding is given by the solution φ = x − t when ψ is a compact coordinate. In thiscase,

(3)ds2 = dudv − u2dψ2, (6.4)

where u ≡ t−x and v ≡ t+x represent null coordinates and ψ is identified with ψ+L forsome arbitrary constant L. A linear translation on ψ corresponds to a null boost. Since ψis compact, the geometry of a constant x surface, with line–element ds2 = dt2−(x−t)2dψ2,is given by R×S1. This surface resembles a lorentzian cone, because there exists a vertexat x = t. Thus, the spacetime (6.4) is geodesically incomplete. It represents a lorentzianorbifold whose vertex moves at the speed of light [35]. In conclusion, therefore, thetopology of the embedding space is determined by the specific boundary conditions thatare chosen when solving Eq. (2.9), as well as the range of values taken by the extracoordinate.

6.2 Embedding of (2 + 1)–dimensional Spaces

Further insight may be gained by considering the embedding of 3–dimensional spaces infour dimensions. As an example, we consider the line–element in (2+1) dimensions givenby

(3)ds2 = dt2 − dρ2 − λ2ρ2dθ2, (6.5)

where λ is a constant and 0 ≤ θ ≤ 2π. When λ = 1− 4mG, this represents the spacetimegenerated by a static point particle of mass m, where G is the gravitational constant

3In the rest of this Section Greek indices take values in the range (0, 1) only.

in (2 + 1) dimensions [36]. As is well known, this space is flat for ρ 6= 0. However, itis not globally Ricci–flat. The non–zero components of the Ricci tensor are (3)Rρ

ρ =(3)Rθ

θ = 2πλ−1(λ − 1)δ(2)(ρ), where δ(2)(ρ) is the Dirac delta function at the originof the 2–surface t = constant, the normalization of which is defined by the condition∫ 2π0 dθ

∫

∞

0 dρδ(2)(ρ)ρ = 1. In fact, Eq. (6.5) represents a conical spacetime which is flateverywhere except at one point corresponding to the vertex ρ = 0. (The constant t sectionsmay be viewed as euclidean planes in which a wedge with opening angle 2π(1− λ) is cutout and its edges identified).

Now, following Campbell’s method, the simplest embedding of metric (6.5) for ρ 6= 0in a (3 + 1)–dimensional, Ricci–flat spacetime is given by

(4)ds2 = dt2 − dρ2 − λ2ρ2dθ2 − dψ2. (6.6)

If ψ is a non–compact coordinate (−∞ ≤ ψ ≤ +∞), Eq. (6.6) is the metric of a static,vacuum cosmic string, where µ ≡ Gm/G is the linear mass density lying on the ψ–axisand G is the gravitational constant in (3 + 1) dimensions [37, 38]. This spacetime is flatfor ρ 6= 0 and the surfaces defined by t = constant and ψ = constant have the sametopology as a cone. The Ricci components are (4)Rρ

ρ = (4)Rθθ = 2πλ−1(λ− 1)δ(2)(ρ).

This embedding (6.6) is not global, however, because the embedding space is onlylocally Ricci–flat. On the other hand, the space defined by Eq. (6.5) may be globallyembedded in (3 + 1)–dimensional Minkowski spacetime. If we define a new coordinate

z ≡ (1 − λ2)1/2ρ, (6.7)

Eq. (6.5) transforms to

(3)ds2 = dt2 − 1

1 − λ2dz2 − λ2z2

1 − λ2dθ2. (6.8)

Equation (6.8) represents the induced metric on the ξ = 0 hypersurface of the (3 + 1)–dimensional spacetime

(4)ds2 = dt2 − λ2

1 − λ2dξ2 − λ2

1 − λ2(ξ + z)2dθ2 − 1

1 − λ2dz2 − 2λ2

1 − λ2dξdz (6.9)

and the coordinate transformation

ξ =

√1 − λ2

λρ− z (6.10)

maps this metric onto Minkowski space.We conclude, therefore, that the application of Campbell’s theorem allows us to embed

the spacetime (6.5) into (6.6) locally, but not globally, in the sense that all points on thespacetime (6.5) are included in the embedding. Since the spacetimes represented by Eqs.(6.5) and (6.6) have a conical singularity at the point ρ = 0, Campbell’s theorem willonly work for ρ 6= 0. The reason for this restriction is that Campbell’s theorem assumesimplicitly that the extra coordinate vector ∂/∂ξ is orthogonal to Eq. (6.5), as can beseen from the general expression (2.2). A global embedding of Eq. (6.5) in Minkowskispace may be achieved, however, by dropping this restriction. In this case, the embeddingspacetime does not inherit the topological defect of the lower–dimensional manifold.

7 Discussion and Conclusions

We have employed Campbell’s embedding theorem in a number of settings. Firstly, weconsidered spacetimes for which the Killing vector is covariantly constant. This classincludes a number of physically interesting spaces, such as the electromagnetic and gravi-tational plane waves, as well as the more general plane–fronted waves. Although we foundembedding spaces for waves with arbitrary amplitude, these embeddings could in princi-ple be generalized by finding new solutions to Eqs. (2.7) and (2.8). We also consideredspacetimes in which the Killing vector is not covariantly constant, including those whichare solutions to vacuum GR such as the stationary van Stockum solutions. An embeddingfor the general class of n–dimensional Einstein spaces was found and we also discussedthe local and global embedding of some lower–dimensional spaces.

Campbell’s theorem is closely related to Wesson’s interpretation of 5–dimensional,vacuum Einstein gravity [8, 9, 10]. In view of this, it would be of interest to consider theembedding of 4–dimensional, cosmological solutions in 5–dimensional, Ricci–flat spaces.For example, inflationary cosmology is thought to be relevant to the physics of the veryearly Universe [39, 40]. During inflation, the scale factor of the Universe accelerates andthis acceleration is driven by the potential energy associated with the self–interactionsof a scalar field. Different inflationary solutions correspond to different functional formsfor the potential of this field. However, Campbell’s theorem implies that all inflation-ary solutions can also be generated, at least in principle, from 5–dimensional, vacuumEinstein gravity. This implies the existence of a correspondence between inflationary cos-mology and Einstein’s theory in five dimensions. In principle, such a relationship couldbe formulated by employing Campbell’s theorem.

Although Campbell’s theorem relates n–dimensional theories to vacuum (n + 1)–dimensional theories, it does not establish a strict equivalence between them. It is there-fore important to determine when such theories are equivalent. Clearly, this is a moresevere restriction than embedability. Two notions of equivalence that could be consideredare dynamical equivalence and geodesic equivalence. Dynamical equivalence would implythat the dynamics of vacuum n-dimensional theories is included in the vacuum (n + 1)-dimensional theories. In that case, the embedding would be given by Eq. (2.2) with φ = 1.This would then imply that (n+1)Rαβ = (n)Rαβ = 0 and that (n+1)Rαn = (n+1)Rnn = 0 [25].

Alternatively, one may consider geodesic equivalence, in the sense of Mashhoon et al.

[41]. In this case the (3 + 1) geodesic equation induces a (2 + 1) geodesic equation plus aforce term F α:

d2xα

ds2+ Γαβγ

dxβ

ds

dxγ

ds= F α. (7.1)

For geodesic equivalence one would therefore require F α = 0, which is clearly so when

∂gαβ∂ψ

= 0. (7.2)

It would be interesting to ask whether these equivalences hold in more general settings.

Acknowledgments JEL was supported by the Particle Physics and Astronomy Re-search Council (PPARC), UK. CR was supported by CNPq (Brazil). RT benefited from

SERC UK Grant No. H09454. SR was supported by a PPARC studentship. CR thanksthe School of Mathematical Sciences for hospitality where part of this work was carriedout. We thank Roustam Zalaletdinov for discussions. We would also like to thank thereferees for helpful comments.

References

[1] Campbell J E 1926 A Course of Differential Geometry (Oxford: Clarendon Press)

[2] Kaluza T 1921 Preus. Acad. Wiss. K1 966Klein O 1926 Zeit. Phys. 37 895Klein O 1926 Nature 118 516

[3] Appelquist T, Chodos A and Freund P G O 1987 Modern Kaluza–Klein Theories

(New York: Addison–Wesley)

[4] Duff M J, Nilsson E W and Pope C N 1986 Phys. Rep. 130 1

[5] Witten E 1981 Nucl. Phys. B186 412Duff M J 1994 Kaluza–Klein Theories in Perspective, Preprint NI–94–015, hep–th/9410046Maia M D 1995 Hypersurfaces of Five Dimensional Vacuum Spacetimes, gr–qc/9512002

[6] Green M B, Schwarz J H and Witten E 1988 Superstring Theory (Cambridge: Cam-bridge University Press)

[7] Kolb E W and Turner M S 1990 The Early Universe (New York: Addison–Wesley)

[8] Wesson P S and Ponce de Leon J 1992 J. Math. Phys. 33 3883Wesson P S 1992 Phys. Lett. B276 299Liu H and Wesson P S 1992 J. Math. Phys. 33 3888Wesson P S 1992 Mod. Phys. Lett. A7 921

[9] Wesson P S 1992 Astrophys. J. 394 19

[10] Billyard A and Wesson P S 1996 Gen. Rel. Grav. 28 129

[11] Witten E 1988 Nucl. Phys. B311 46Mandal G, Sengupta A M and Wadia S R 1991 Mod. Phys. Lett. A6 1685Witten E 1991 Phys. Rev. D44 1991Mann R B 1992 Gen. Rel. Grav. 24 433Callan C G, Giddings S B, Harvey J A and Strominger A 1992 Phys. Rev. D45

R1005Russo J G, Susskind L and Thorlacius L 1992 Phys. Rev. D46 3444Russo J G, Susskind L and Thorlacius L 1993 Phys. Rev. D47 533Mann R B and Ross S F 1993 Phys. Rev. D47 3312Gegenberg J and Kunstatter G 1993 Phys. Rev. D47 R4192Mazzitelli F D and Russo J G 1993 Phys. Rev. D47 4490

Perry M J and Teo E 1993 Phys. Rev. Lett. 71 2669Onder M and Tucker R W 1994 Class. Quantum Grav. 11 1243Navarro–Salas J, Navarro M, Talavera C F and Aldaya V 1994 Phys. Rev. 50 901Mignemi S 1994 Phys. Rev. D50 R4733Lidsey J E 1995 Phys. Rev. D51 1829Tseytlin A A 1995 Class. Quantum Grav. 12 2365

[12] Rippl S, Romero C and Tavakol R 1995 Class. Quantum Grav. 12 2411

[13] Verbin Y 1994 Phys. Rev. D50 7318

[14] Giddings S, Abbott J and Kuchar K 1984 Gen. Rel. Grav. 16 751

[15] Kramer D, Stephani H, MacCallum M and Herlt E 1980 Exact Solutions of Einstein’s

Field Equations (Cambridge: Cambridge University Press)

[16] Fronsdal C 1959 Phys. Rev. 116 778

[17] Eisenhart L P 1949 Riemannian Geometry (New Jersey: Princeton University Press)

[18] Friedman A 1961 J. Math. Mech. 10 625Friedman A 1965 Rev. Mod. Phys. 37 201

[19] Schouten J A and Struik D J 1921 Am. J. Math. 43 213

[20] Kasner E 1921 Am. J. Math. 43 126Kasner E 1921 Am. J. Math. 43 130

[21] See for example Misner C W, Thorne, K S and Wheeler, J A 1973 Gravitation, W.H.Freeman and Company

[22] See for example Wald R M 1984, General Relativity, The University of Chicago Press.

[23] Magaard L 1963, Zur einbettung riemannscher raume in Einstein-raume und konfor-

euclidische raume, PhD thesis, (Kiel)

[24] Goenner H 1980 In General Relativity and Gravitation One Hundred Years after the

Birth of Albert Einstein Vol I, p. 441 (New York: Plenum Press)

[25] Romero C, Tavakol R and Zalaletdinov R 1996 Gen. Rel. Grav. 28 365

[26] Brinkman H W 1925 Math. Ann. 94 119

[27] Baldwin O R and Jeffery G B 1926 Proc. Roy. Soc. Lond. A111 95

[28] Penrose R 1965 Rev. Mod. Phys. 37 215

[29] Guven R 1987 Phys. Lett. B191 275Amati D and Klimcik C 1989 Phys. Lett. B219 443

[30] Horowitz G T and Steif A R 1990 Phys. Rev. Lett. 64 260Horowitz G T and Steif A R 1990 Phys. Rev. D42 1950

[31] Brinkman H W 1923 Proc. Natl. Acad. Sci. 9 1

[32] Friedan D 1980 Phys. Rev. Lett. 45 1057Jack I, Jones D and Mohammedi N 1989 Nucl. Phys. B322 431Callan C, Friedan D, Martinec E and Perry M 1985 Nucl. Phys. B262 593

[33] Hawking S W and Ellis G F R 1973 The Large Scale Structure of Spacetime (Cam-bridge: Cambridge University Press)

[34] Clarke C J S 1970 Proc. R. Soc. Lond. A314 417

[35] Horowitz G T and Steif A R 1991 Phys. Lett. B258 91

[36] Deser S, Jackiw R and t’Hooft G 1984 Ann. Phys. 152 220

[37] Vilenkin A 1981 Phys. Rev. D23 852

[38] Hiscock W A 1985 Phys. Rev. D31 3288

[39] Starobinsky A A 1980 Phys. Lett. B91 99Guth A H 1981 Phys. Rev. D23 347Sato K 1981 Mon. Not. R. astron. Soc. 195 467Linde A D 1982 Phys. Lett. B108 389Albrecht A and Steinhardt P J 1982 Phys. Rev. Lett. 48 1220

[40] Olive K A 1990 Phys. Rep. 190 307Liddle A R and Lyth D H 1993 Phys. Rep. 231 1

[41] Mashhoon B, Liu H and Wesson P S 1994 Phys. Lett. B331 305